1.1
Three standard methods
- Factorisation (when it factors nicely).
- Completing the square (turns ax²+bx+c into a perfect square).
- Quadratic formula x = (−b ± √(b²−4ac)) / (2a).
Aatish - Lesson 1 Quadratic and Simultaneous Equation
1. How to find roots of a quadratic
1.2
Worked examples for each method
- Compare the steps side‑by‑side.
- Always rewrite into ax² + bx + c = 0 first.
2. Solving a system that includes a quadratic
2.1
Simultaneous equations
- Use substitution to reduce to a single quadratic.
- Factorise to get possible values and check both.
3. Parabola essentials
3.1 What y = ax² + bx + c looks like • 3.2 When a > 0 • 3.3 When a < 0 • 3.4 Vertex form when a > 0 • 3.5 Vertex form when a < 0
3.1
What y = ax² + bx + c looks like
- If a>0 the curve opens upwards; if a<0 it opens downwards.
- The lowest/highest point is called the vertex (turning point).
3.2
When a > 0
- Parabola has a minimum at the vertex.
- The axis of symmetry passes through the vertex.
3.3
When a < 0
- Parabola has a maximum at the vertex.
- Same symmetry idea but curve opens down.
3.4
Vertex form when a > 0
- Write as a(x+p)² + q; vertex is (−p, q).
- q is the minimum value.
3.5
Vertex form when a < 0
- Write as a(x+p)² + q; vertex is (−p, q).
- q is the maximum value.
4. Graphing worked examples
4.1
Example 1: x‑intercepts and range
- Find where y=0 to get intercepts.
- Use symmetry to locate the vertex and state the range.
4.2
Example 2: rewrite to vertex form
- Complete the square to read off the maximum point.
- Then solve for intercepts if needed.
5. Absolute value of a quadratic
5.1
Sketch of y = |x² + 4x − 12|
- Reflect any part below the x‑axis upwards.
- Vertices occur where the original quadratic meets the x‑axis.
5.2
How many solutions for |x² + 4x − 12| = k
- Draw a horizontal line y=k and count intersections.
- Special case at the peak/trough gives 3 solutions.
6. Discriminant and types of roots
6.1
Summary: b² − 4ac
- >0 two distinct real roots; =0 one repeated root; <0 no real roots.
- Use this to predict how the graph meets the x‑axis.
6.2
How the graph meets the x‑axis
- a>0 or a<0 cases shown for intersect/touch/miss.
- Connect each picture to the discriminant condition.
7. Straight line and parabola
7.1
Intersect vs tangent vs no meet
- Two points ⇒ b²−4ac>0; tangent ⇒ =0; no meet ⇒ <0.
7.2
Finding a tangent
- Set up the equations together and apply b²−4ac=0 to solve the parameter.
8. Parameter problems using the discriminant
8.1 No real roots: find the allowed q • 8.2 Real roots exist: find k • 8.3 Equal roots: find h • 8.4 Touches x‑axis once: find p • 8.5 Does not meet x‑axis: range of k • 8.6 Intersects at two points: range of m
8.1
No real roots: find the allowed q
- Rewrite to ax²+bx+c=0 and apply b²−4ac<0.
8.2
Real roots exist: find k
- Apply b²−4ac ≥ 0 and be careful when dividing by a negative number.
8.3
Equal roots: find h
- Use b²−4ac = 0 to get a repeated root (touches x‑axis).
8.4
Touches x‑axis once: find p
- Again b²−4ac = 0; expand and solve the resulting equation in p.
8.5
Does not meet x‑axis: range of k
- For a>0, the whole graph is above x‑axis when b²−4ac < 0.
8.6
Intersects at two points: range of m
- Combine line and parabola into one quadratic in x; require b²−4ac>0.
9. Quadratic inequalities
9.1
< 0 means ‘below the x‑axis’
- Solve the quadratic to get roots, then shade the region below the axis.
- Answer lies strictly between the two roots.
9.2
≥ 0 means ‘on/above the x‑axis’
- Outside the roots (or touching if discriminant=0).
- Include the endpoints because of the ‘=’.
